Question

3. I purchase parts from two different suppliers, A and B.  Testing the products they have sent I found that 35 of 80 parts from A are defective and 66 of 80 parts from B are defective.  Test at the 5% significance level that p_A=p_B
   1. Should you use a t or z test?

2. State the null hypothesis

3. Compute the test statistic

4. What is the confidence interval for p_A-p_B

Answer 1

a)
z -test

b)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

c)
p1cap = X1/N1 = 35/80 = 0.4375
p1cap = X2/N2 = 66/80 = 0.825
pcap = (X1 + X2)/(N1 + N2) = (35+66)/(80+80) = 0.6313

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.4375-0.825)/sqrt(0.6313*(1-0.6313)*(1/80 + 1/80))
z = -5.08

d)
Here, , n1 = 80 , n2 = 80
p1cap = 0.4375 , p2cap = 0.825

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.4375 * (1-0.4375)/80 + 0.825*(1-0.825)/80)
SE = 0.0699

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.4375 - 0.825 - 1.96*0.0699, 0.4375 - 0.825 + 1.96*0.0699)
CI = (-0.5245 , -0.2505)